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Theorem ifcldadc 3400
Description: Conditional closure. (Contributed by Jim Kingdon, 11-Jan-2022.)
Hypotheses
Ref Expression
ifcldadc.1  |-  ( (
ph  /\  ps )  ->  A  e.  C )
ifcldadc.2  |-  ( (
ph  /\  -.  ps )  ->  B  e.  C )
ifcldadc.dc  |-  ( ph  -> DECID  ps )
Assertion
Ref Expression
ifcldadc  |-  ( ph  ->  if ( ps ,  A ,  B )  e.  C )

Proof of Theorem ifcldadc
StepHypRef Expression
1 iftrue 3378 . . . 4  |-  ( ps 
->  if ( ps ,  A ,  B )  =  A )
21adantl 271 . . 3  |-  ( (
ph  /\  ps )  ->  if ( ps ,  A ,  B )  =  A )
3 ifcldadc.1 . . 3  |-  ( (
ph  /\  ps )  ->  A  e.  C )
42, 3eqeltrd 2159 . 2  |-  ( (
ph  /\  ps )  ->  if ( ps ,  A ,  B )  e.  C )
5 iffalse 3381 . . . 4  |-  ( -. 
ps  ->  if ( ps ,  A ,  B
)  =  B )
65adantl 271 . . 3  |-  ( (
ph  /\  -.  ps )  ->  if ( ps ,  A ,  B )  =  B )
7 ifcldadc.2 . . 3  |-  ( (
ph  /\  -.  ps )  ->  B  e.  C )
86, 7eqeltrd 2159 . 2  |-  ( (
ph  /\  -.  ps )  ->  if ( ps ,  A ,  B )  e.  C )
9 ifcldadc.dc . . 3  |-  ( ph  -> DECID  ps )
10 exmiddc 778 . . 3  |-  (DECID  ps  ->  ( ps  \/  -.  ps ) )
119, 10syl 14 . 2  |-  ( ph  ->  ( ps  \/  -.  ps ) )
124, 8, 11mpjaodan 745 1  |-  ( ph  ->  if ( ps ,  A ,  B )  e.  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    \/ wo 662  DECID wdc 776    = wceq 1285    e. wcel 1434   ifcif 3373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-dc 777  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-if 3374
This theorem is referenced by:  updjudhf  6677  eucalgval2  10815  lcmval  10825
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